NASA - Apps.dtic.mil
NASATechnicalAD-A242 332IPaper3096."AVSCOMTechnicalReport91 -C-020August 1991A Method for DeterminingSpiral-Bevel Gear ToothGeometry for FiniteElement AnalysisRobert F. Handschuhand Faydor L. LitvinUS AR M YAVIATIONSYSTEMS COMMANDAVIA11ON,T'"I*qnIlmJftdViA(I;V'.,NASA91- 74967I:I I '.IIl I .lri:I;rIU
NASATechnicalPaper3096AVSCOMTechnicalReport91 -C-0201991A Method for DeterminingSpiral-Bevel Gear ToothGeometry for FiniteElement AnalysisRobert F. HandschuhPropulsion DirectorateU.S. Army-AVSCOMLewis Research CenterCleveland, OhioFaydor L. LitvinUniversity of Illinois at ChicagoChicago, Illinois-Tri 0TIEL.s,.Istribkti /Cgoqg- ive i 1adorAvailabtityDistNational Aeronautics andSpace AdministrationOffice of ManagementScientific and TechnicalInformation Program.Speeial
SummaryAn analytical method has been developed to determine geartooth surface coordinates of face-milled spiral bevel gears. Themethod uses the basic gear design parameters in conjunctionwith the kinematical aspects of spiral bevel gear manufacturingmachinery. A computer program entitled 'SURFACE- wasdeveloped to calculate the surface coordinates and providethree-dimensional model data that can be used for finiteclement analysis. Development of the modeling method andan example case are presented in this report. This method ofanal. sis could also be applied in gear inspection and near-netshape gear forging die design.IntroductionSpiral bevel -,cars are currently used in all helicopter powertransmission systems. This type of gear is required to turn the-cncr typecorner from a horii.ontal engine to the vertical rotor shaft.These (,ears carry large loads and operate at high rotationalspeeds. Recent research has focused on understanding manysaspects of spiral bevel gear operation, including gear geometrv(refs. I to 12). gear dynamics (refs. 13 to 15). lubrication(ref. 16). stress analysis and measurement (refs. 17 to 21).misali-nment (refs. 22 and 23). and coordinate measurements(refs. 24 and 25). as well as other areas.Research in gear geometry hits concentrated onunderstanding the meshing C,action of spiral be,,el gears (rels. 8to II). This meshing action often results in much vibrationand noise due to an inherent lack of conjugation. Vibrationstudies (ref. 26) have shown that in the frequency spectrumofan entire helicopter transmission, the highest responsecanZ71be that from the spiral bevel gear mesh. Therefore if noisereduction techniques Lire to he imiplemiented effectively. themeshing action of spiral nevel gears must be understood.Also. investigators (refs. 18 and 19) have found that typicaldesign stress indices for spiral bevel gears can be significant]ydifferent from thoseCmeasured experimentall,,. In addition tomaking the dcsign process one of trial and error (forcing oneto rclyti past experience). this incotnsistencvmakesextrapolating oxera ,,ide range ofesizes difficult. and an overlycon,,crx ati, c design can result.Research has been ongoing in an attenmpt to predict stresseswi.e . hending and contact) bx using the finite clement method.A grcat deal of \ýork (rels. 27 tw 30) has g-,one into finiteelment modeling of parallel axis ,ears to determine the stressfield. Loads are typically applied at the point of highest singletooth contact, and then the stress in the fillet re ion isexamined. Computer programs that perform this type ofanalysis are usually twAo dimensional in nature and ha\ecomputer storage requirements that are small enough forpersonal computers. These attributes make them very popularand attractive to designers. However, a limited number ofresearchers (refs. 16 and 21) have in\estigated finite elementanalysis of spiral bevel gears.Parallel axis components (involute tooth geometr\ ) ha\ eclosed-form solutions that determine surface coordinates.These coordinates can be used as input to finite elementmnethods and other analysis tools. Spiral bevel gears. on theother hand, do not have a closed-fornm solution to describe theirsurface coordinates. Coordinate locations must be solvednumerically. This process is accomplished by modeling thekinematics of the cutting or grindingz machinerx and the-baicLardaimetry of the basic gear design.The objective of the research reported herein was to de\elopa method for calculating spiral bevel gear-tooth surfacecoordinates and a three-dimensionalb model forfinite elementaAccomplishment of this task required a basicunderstanding of the gear manufacturing process. \%hich isdescribed herein by use of differential geometr.n technique,,d rh(ref. I). Both the rmanufacturitin machine settings and the basicgear design data were used in a numerical anal\sis procedurethatyielded the toothsurfacecoordinates.After the toothufcs(rvnosidsIwrecieatresurfaces (drive and coast sides) ,.'ere described. at three-deonamo For the thsembledA ompterprogram. SURFACE, was developed to automate thc"calculation of the tooth surface coordinates, and hence, thecoordinates fo r the gear-tooth three-dimensional finite elementmodel. The development of the analytical model is explained.and an example of he finite element method is presented.Determination of Tooth SurfaceCoordinatesThe spiral gear machining process describc,d in this paper isthat of heftace-milled tx pc. Spiral bevel gears, man, lctured inthis wax, are used Cxtensixe,,l in aerospace porxcr transnissions(i.e. helicopter imain'tail rotor transm,,isions) to transmit ploscrbctx\ecn hori/ontal gas turbine engines and the \crtlical rooir
shaft. Because spiral bevel gears can accommodate various shaftorientations, they allow greater freedom for overall aircraftlayout.In the tollowing sections the methox of determining gear-toxothsurface coxrdinates will be described. The manufacturing processmust first be understood and then analytically described,Equations must be developed that relate machine and workpiecemotions and settings with the basic gear design data. Thesimultaneous solution of these equations must be donenumerically since no closed-form solution exists. A descriptionof this procedure follows,G;ear ManufactureSpiral bevel gears are manufactured on a machine like the oneshown in figure 1. This machine cuts away the material betweenthe concave and convex tooth surfaces of adjacent teethsimultaneously. The machining process is better illustrated inCradle housing\Cradlefigure 2. The head cutter (holding the cutting blades or thegrinding wheel) rotates about its own axis at the proper cuttingspeed. independent of the cradle or workpiece rotation. The headcutter is connected to the cradle through an eccentric that allowsadjustment of the axial distance between the cutter center andcradle (machine) center, and adjustment of the angular positionbetween the two axes to provide the desired mean spiral angle.The cradle and workpiece are connected through a system ofgears and shafts, which controls the ratio of rotational motionbetween the two (ratio of roll). For cutting,. the ratio is constant.but for grinding. it is a variable.Computer numerical controlled (CNC) versions of the cuttingand grinding manufacturingC processes are currently beingdeveloped. The basic kinematics, however, are still maintainedfor the generation process: this is accomplished by the CNCmachinery duplicating the generating motion through point-topoint control of the machining surfhce and location of theworkpiece.\/--Work offset.--Machinecenterto-back.Machin,e/Root angle". .\centerWork baseSliding baseiF iur,.e Ie t,.d to gener atespiral hcel Lcar-rtoothM achinIh,,urlaic
Cradle axis -\,\/Cutter axisXcXra le houisinigAr-Head cutter{ §Wokpiece]ý .ocV-bycf" rcTop vieweInside blade (convex side);u IA1hosnCradleAhouin WorkpieceCradleZBManW " contactxxcOcCr-r"-"- Spiral angle"- Cutter circleMachinecenterAoI\'/Cutter centerFront viewh:Iurc 2.--Orieniation oftOutside blade (concave side);orkpiece to generation mnachiners.Coordinate Transformationsu IABIFigure 3.-Head-cuLtter cone ,urtace,.The surface of a generated gear is an envelope to the familyof surfaces of the head cutter. In simple terms this means thatblades cut the convex and conca,,e sides of the gear teeth.respectively. A point on the cutter blade surface is determinedthe points on the generated tooth surface are points of tangencyby the following:to the cutter surface during manufacture. The conditionsnecessary for envelope existence are given kinematically by theequation of meshing. This equation can be stated as follows: thenormal of the generating surface must be perpendicular to therelative velocity between the cutter and the gear-tooth surfaceat the point in question (ref. 1).The coordinate transformation procedure that will now bedescribed is required to locate any point from the head cutterinto a coordinate system rigidly attached to the gear beingmanufactured. Homogeneous coordinates are used to allowrotation and translation of vectors simply by multiplying thematrix transformations, The method used for the coordinatetransformation can be found in references I. 5, and 8 to II.Let us begin with the head-cutter coordinate system S, shownin figure 3. This report assumes that the cutters are straight sided(not curved as commonly used on the wheel for final grinding),Surface coordinates u and 0 determine the location of a currentpoint on the cutter surface as well as the orientation of the currentpoint with respect to coordinate system S,. Angles ý,/, and ý,,,are the inside and outside blade angles. The inside and outsider cotui sinr iu cos-' si. '3it sin "os 0(1)Lwhere fixed value r i, ie radius of the blade at x, 0. andis the blade angle. : ararneters u and 0 locate a point in systemS, and are unkrowns whose value will be deternined.The head-cutter coordinate system S, is rigidly connected tocoordinate ,)stem S, (fig. 4). System S, is rigidly connected tothe cralle that rotates about the x,,, axis of the machinecoordinate system S,,. Coordinate .;ystem S,, is a fixedcoordinate system and is connected to the machine frame.To reference the head cutter in coordinate system S,. thefollowing transfornation is necessary:
YS YmZl,,zs 0cos 6, sin o,()(04:sin 0,cos 0,((3)Coordinate systemn S, locates the machine center. andcoordinate systemi S ,,orients the pitch apex of' the -cear heingnmanufalctured. The transformiation fromt coordinate s' stenI .5,.tocoordna eS sterilSp, requires the mcietool settinos Land E,, alono wkith dedendumn anefle 6 fromn the comnponentdesig-n (see figs. 5 and 6). Machine tool settingls L,, and E,, canhe found f'romi the summnary sheet that t),picalI\ accompanies aor the methods in reference 8 can he used. Relcrence 8converts, standard machine tool settines lir the slidinu, base. theofftset, and the machine center-to-hack into settings L,, and E,.as showkn iii table 1. The transf'ormation mnatrix is -,i\cnh\o OckCcear.Left-hand member; s OcOsYCzOmpCyRight-hand member, S OcOsTop viev!0 cosq4: sin I/4:F sin1(r(2) SI) coss i (/ No sYp Ya., Yw Ym0oZ0,0j.Th upe2qCoriaesseq is tie cradle. angi-lenSoiisChereand s is the distance between theand lowecr signs preceding the \arious termis in this matrixtranst'Ornnat in ( andl the rest ut the paper) pertain to left- and right hand gecars, respectik elk .No%% . to transfoirm fromn S, to the fixed cooirdinatc systefn Sjthe roll angle of' the cradle (.) is used. This trans hlrrlat ion is"given by4w/Op.a. OwZIP Za,0ALWzm-L mFront viewIirc5(0i1,ilirdtinaic hIirew'\I ,kIilIcniationI'ii, ,nci ýiic i muit haind ,caj mitccj
X,rCos 6-a,O,0- sin"6Lqi60 POP.0m yY OcZWXZa. Z,Top viowYP, Y. YWymi (see fig. 7). This is given byOm co0P,0a.Sp.OS0 Sinl itMOP(5)Wgilcrak' ait',lei- huts) paL rN11tiaL'eTAtti0-Sin it0 Cos J( 0('tullc utigle. ,,,0to Si,,. coordinate transformiation isgvnbt.--StUN CO NVINti( NS OFtNACHINIfN-TO0l. SEir7NGSIEromtret', 8,1Selitntg1/SignRight-thand mneiherCounierelockwike (CCWIClockwisec (C\V)-Clonckwise (CW)Counteclockwise ('C\\')Atboye machtine centerBtelow mactine centerAhove mactine center-Ietow machine centerBcenter-lo-hack.SlIding Nise . Wonrk withdrawal-I/Wonrk advanceWork wiflhdrawalWork advanceWork wididtrtwatViik advanceWork \withdrawalWork advainceA-,j%tln orX,,rid X ,11Lett-harnd nt.',nhcr4-Mauchtining ollsei. E"Vector0the cradle 0,. (this relationship wvillI be described in thle nextsection). Tmiachtine0coordinate System 7,,,,which is fixed to thle component beingmianuflactured. A rotation about the z:,,-axis thiroug h an angle 6,,.is requtired. Angle n/i,,., Shown iti figure. 8. is thle worklpiecerotation angle; it is directly related to thie angle of' rotation of'Front viowI 1,Ic0. C'l(wMIiiC *NýIm oncniialThis is shown in figure 5 fbr a right-hand mecmber and in figure 6lior a lefl-hand member. Figure 7 is given to clarity' thle orientationot' the coordinate Systems and machine tool settings (I11.11,1).Thle next transformation involves rotation of' systemi .S,to S,.Thle common origin for coordinate system S,, and S,, locates theapex of the geatr under consideration with respect to Coordinatesystemn S,,,. This requires rotation about v,, by the pitch angleX511. -, XIA(f.-Xg.-XIcl. Nff ;.Xjfwt,: ,.4 v,
xmXaZOa, OwOP ,,. za, ZwPlane nYmnOypYa(a)XaXPPlane 7cYw\--PlanetI'Xw.OWZP"bi-wZa,0 :a.Owz,Yw(b)yw Z(a) Machine ,ettings, and orientation.Yw(hi Plane 7r and orientation of generated gear coKrdinate,,.YaFigure 7. -()nenLation o" machine settingrand generated gear coordinate ,,yser,,.cos 'p,, sin 0,,00cosO,,00000I000001T s inLeft-hand memberXa xw(OWYa, Yp Right-hand me.inberFigure 8.-Rotation of .orkpiece for left- and right-hand gears during hoothssurface generation.(6)Tooth Surface Coordinate Solution ProcedureUstng these matrix transformations, we can determine thecoordinates in S,, of a point on the generating surface fromIn order to solve for the coordinates of a spiral bevel geartooth surface, the following items must be used simultaneously:the transformation process, the equation of meshing. and thebasic gear design information. The transformation processdescribed previously is used to determine the location of a pointon the head cutter in coordinate system S . Since there are(7)three unknown quantities (u, 0. and ,), three equationsrelating them must be developed.Values for u. 0, and 0, are used to satisfy the equation ofr,, IMIMII,,IIM,,,,,IIM,,,,I IM,Ir,meshing given by references I and 9:orr" IM,,,fl, ) IIM,",, [M,,,I IM,,,, (, )1IM,, Ir, (u,0)(8)n V 0(10)orr, r, (u.0,,)(9)This transformation describes the location of a point in thegear fixed coordinate system based on machine settings (L,,,E,,. q. s. r. and ý). parameters (u. 0. and 0,), and geardesign information (, and 6).6where n is the normal vector to the cutter and workpiecesurfaces at the specified location of interest, and V is therelative velocity between the cutter and workpiece surfacesat the specified location. From the reference 9 equation ofmeshing for straight-sided cutters with a constant ratio of rollbetween the cutter and workpiece, equation (10) is defined as
(1i - r cot V/cos k)cos It sin rXa, XW sI(mn-y)cosý sin0 Fcos -y sin ý sin (q-sinX,)Pp-CXmSPoint P E,, (cossinf- sin I coscos r)zp-Lsin-ycosý sin 0(11)Zmrwhere -y is the root angle of the component beingmanufactured, andT(0 :F q (12))andView AYwn,,,where in,,, is the ratio of angular velocity of the cradle to thatof the workpiece. Since the ratio of roll in this report isassumed to be constant, equation (13) can be written asWO)\XaView AFigure 9.-Orientation of gear io be generaled. ,Aih a,,unied po,,ition, randThe axial position must match the value found fromtransforming the cutter coordinates S, to workpiececoordinates S . This is satisfied by the following (fig. 9):andLOt,w13 )((-Ya do(, do,(itdt(16)Z,, - z 0orthereforeFinally the radial location from the work axis of rotation mustbe satisfied. This is accomplished by using the magnitude ofthe location in question in the x,,-v,, plane (see fig. 9):tuor I(14)In,.,,,. - (x;, Equation (14) is the relationship between the cradle andworkpiece for a constant ratio of roll and is used directly inequation (6).Gear design information is then used to establish anallowable range of values of the radial (r) and axial (z)positions that are known to exist on the gear being generated.This is shown in figure 9.First the equation of meshing must be satisfied. This wasshown earlier to ben -V( 17 )0 ,) 0f 2(u , O.,It 0or/] (u, 6, O,) 0(15)vy;,(V0, 0, 00(18)(19)Now a system of three equations (eqs. (15). (17). and (19))is s ys te offor the t e p1 m t. u,7). andis solved simultaneously for the three parameters u, . and0,. for a given gear design with a set of machine toolsettings. These are nonlinear algebraic equations that can besolved numerically with commercially available mathematicalsubroutines. These equations are then solved simultaneouslyfor each location of interest along the tooth flank, as shownin figure 10. In the SURFACE program a 10 by 10 grid ofpoints is used on each side of the tooth. From the surface grids.the active profile (working depth) occupied by a single toothis defined.7
WholeConvex sideCl aranceDededum Dedendum10 Points\ (10 Points ),-Points- -DedendumFhienr 10)--Calculation points (1 h%)1 0g rid,. i.e. 101)points each side)tor co ncas e and ci ns e \iISiesfi tooth su rt ace.Application of Solution TechniqueAn application of the techniques previously discussed willbe presented. The component to be modeled was fromthe NASA Lewis Spiral Bevel Geai Test Facility. A photog raph of the spiral bevel gear mesh is shown in figure II., andthe design data for the pinion member are shown in table 11.zrThe gear design data were used along with the methods ofreference 9 to determine the machine tool settinos for straiahtsided cutters (see table 11). These values were then used asinput to SURFACE. This program calculates the coordinatesflo\XTABLE II.C74I-NSprlbee(a) Pinion dcsiszn dataNumber of teethpinion . . t12,cear . 36Dedendurn angle. d g . . .1.0Addendum anglc. dc . . 3.883Pitch anule. deg . . . . . . . . . . . . . . . . . . . . . . . 18.43390 .0Shaft am! Ic. dcM ean spiral angle. deg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 .[-ace wkidth. mmrr(in.) .- I. . .25.4 (l.0tMean cone distance. mmn (in.)ý. . 81.05 (3.191)Inside radiusJ o) gear blank. mmn (in.) .-. 15.3 o),6094)). . . . 2.0 )32 (0 . 1)80. . . .o p land thickness. mm(in .) . . .T.762 (0.t0301. (.(.Cl earance. mmi (in.) .bt) Generation machine settinesf Concaxgncipretof the :oncave and convex sides of the gear tooth (fig. 10).orients the surfaces such that the top land is of the properwidth, and then generates the required data for the three-EXAMPLE CASE OF SURFACE COORDINATE GENERATIONRadius of cutter. i. miin (in.)Blade anele. ,'. deevector sumi. I.,,. tmi (in.)Machine offset. I6,. min ( in.)Cradle to cutter distance. s. nun (in.)Cradle angle. (1, &LRatio of roll. miInitial cutter lengtht. it. turti (in.Initial Cutter orientatiin. (4, de,Initial cradll irientation, o, . (lei!erone75.2212 (2.96 15)78.,1329 (3.0761i161.3582.31.0)363 ((1.04(18)-1.4249) -0.01561i13.9802, 0. 1567)-4,4856 I-0(.1 706674,839 (2.9646)71, 42.(5)64.0)153.822t.O 31:8462(1.3217672395 9.43iI81. 1 (7. 131120)).)11j()
dimensional modeling program PATRAN (ref. 31). The detailsS of the procedure are described in the following paragraphs.\ calculation gridgrd\-HeelXt-urfaXmSurface C'oordinate CalculationToUsing figures 10 and 12 as retfrences, we will describe thecalculation procedure for surface c,,ordinates. First. theconcave side of the tooth is completely defined before mov ingto the convex side. These points are calculated hy starti
a method for calculating spiral bevel gear-tooth surface speeds. Recent research has focused on understanding manys b - coordinates and a three-dimensional model for finite element aspects of spiral bevel gear operation, including gear geometrv (refs. I to 12). gear dynamics (refs. 13 to 15).
mil-dtl-22992 mil-dtl-24308 mil-dtl-25516 mil-dtl-25955 mil-dtl-26482 mil-dtl-26500 mil-dtl-26518 mil-dtl-27599 mil-dtl-28731 mil-dtl-28748 mil-dtl-28840 mil-dtl-32139 mil-dtl-38999 mil-dtl-39024 mil-dtl-3933 mil-dtl-5015 mil-dtl-5015 h mil-dtl-55116 mil-dtl-55181 mil-dtl-55302 mil-dtl-83413 mil-dtl-83503 mil-dtl-83513 mil-dtl
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